Octave 3.8, jcobi/2

Percentage Accurate: 62.8% → 97.6%

Time: 25.5s

Alternatives: 10

Speedup: 9.5×

Specification

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Initial Program: 62.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Alternative 1: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(\left(2 \cdot \frac{\beta}{\alpha \cdot \alpha} - \frac{\beta + 2}{\frac{\alpha \cdot \alpha}{2 + \beta \cdot 2}}\right) + \left(\frac{2 + 2 \cdot i}{\alpha} + \frac{\beta + 2 \cdot i}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{t_1} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.9998)
     (/
      (+
       (/ beta alpha)
       (+
        (-
         (* 2.0 (/ beta (* alpha alpha)))
         (/ (+ beta 2.0) (/ (* alpha alpha) (+ 2.0 (* beta 2.0)))))
        (+ (/ (+ 2.0 (* 2.0 i)) alpha) (/ (+ beta (* 2.0 i)) alpha))))
      2.0)
     (/
      (+
       (/ (* (- beta alpha) (/ (+ alpha beta) (+ beta (fma 2.0 i alpha)))) t_1)
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.9998) {
		tmp = ((beta / alpha) + (((2.0 * (beta / (alpha * alpha))) - ((beta + 2.0) / ((alpha * alpha) / (2.0 + (beta * 2.0))))) + (((2.0 + (2.0 * i)) / alpha) + ((beta + (2.0 * i)) / alpha)))) / 2.0;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / (beta + fma(2.0, i, alpha)))) / t_1) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.9998)
		tmp = Float64(Float64(Float64(beta / alpha) + Float64(Float64(Float64(2.0 * Float64(beta / Float64(alpha * alpha))) - Float64(Float64(beta + 2.0) / Float64(Float64(alpha * alpha) / Float64(2.0 + Float64(beta * 2.0))))) + Float64(Float64(Float64(2.0 + Float64(2.0 * i)) / alpha) + Float64(Float64(beta + Float64(2.0 * i)) / alpha)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / Float64(beta + fma(2.0, i, alpha)))) / t_1) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.9998], N[(N[(N[(beta / alpha), $MachinePrecision] + N[(N[(N[(2.0 * N[(beta / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(beta + 2.0), $MachinePrecision] / N[(N[(alpha * alpha), $MachinePrecision] / N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.99980000000000002

   1. Initial program 2.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 20.5%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]

   4. Taylor expanded in beta around 0 20.5%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]

   5. Taylor expanded in alpha around -inf 75.3%

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\alpha} + \left(\frac{\left(2 + 2 \cdot i\right) \cdot \left(\beta + 2 \cdot i\right)}{{\alpha}^{2}} + \frac{\left(-1 \cdot \left(2 + 2 \cdot i\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta - \left(-1 \cdot \left(2 + 2 \cdot i\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right)\right)}{{\alpha}^{2}}\right)\right) - \left(-1 \cdot \frac{2 + 2 \cdot i}{\alpha} + -1 \cdot \frac{\beta + 2 \cdot i}{\alpha}\right)}}{2} \]
   6. Step-by-step derivation

      1. associate--l+ 75.3%

         \[\leadsto \frac{\color{blue}{\frac{\beta}{\alpha} + \left(\left(\frac{\left(2 + 2 \cdot i\right) \cdot \left(\beta + 2 \cdot i\right)}{{\alpha}^{2}} + \frac{\left(-1 \cdot \left(2 + 2 \cdot i\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta - \left(-1 \cdot \left(2 + 2 \cdot i\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right)\right)}{{\alpha}^{2}}\right) - \left(-1 \cdot \frac{2 + 2 \cdot i}{\alpha} + -1 \cdot \frac{\beta + 2 \cdot i}{\alpha}\right)\right)}}{2} \]

   7. Simplified 86.6%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\alpha} + \left(\left(\frac{2 + 2 \cdot i}{\frac{\alpha \cdot \alpha}{2 \cdot i + \beta}} + \frac{\left(-\left(2 \cdot i + \beta\right)\right) - \left(2 + 2 \cdot i\right)}{\frac{\alpha \cdot \alpha}{\beta - \left(\left(-\left(2 \cdot i + \beta\right)\right) - \left(2 + 2 \cdot i\right)\right)}}\right) - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}}{2} \]

   8. Taylor expanded in i around 0 84.2%

      \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\color{blue}{\left(-1 \cdot \frac{\left(2 + \beta\right) \cdot \left(2 + 2 \cdot \beta\right)}{{\alpha}^{2}} + 2 \cdot \frac{\beta}{{\alpha}^{2}}\right)} - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]
   9. Step-by-step derivation

      1. +-commutative 84.2%

         \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\color{blue}{\left(2 \cdot \frac{\beta}{{\alpha}^{2}} + -1 \cdot \frac{\left(2 + \beta\right) \cdot \left(2 + 2 \cdot \beta\right)}{{\alpha}^{2}}\right)} - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

      2. mul-1-neg 84.2%

         \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\left(2 \cdot \frac{\beta}{{\alpha}^{2}} + \color{blue}{\left(-\frac{\left(2 + \beta\right) \cdot \left(2 + 2 \cdot \beta\right)}{{\alpha}^{2}}\right)}\right) - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

      3. unsub-neg 84.2%

         \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\color{blue}{\left(2 \cdot \frac{\beta}{{\alpha}^{2}} - \frac{\left(2 + \beta\right) \cdot \left(2 + 2 \cdot \beta\right)}{{\alpha}^{2}}\right)} - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

      4. unpow2 84.2%

         \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\left(2 \cdot \frac{\beta}{\color{blue}{\alpha \cdot \alpha}} - \frac{\left(2 + \beta\right) \cdot \left(2 + 2 \cdot \beta\right)}{{\alpha}^{2}}\right) - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

      5. associate-/l* 86.6%

         \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\left(2 \cdot \frac{\beta}{\alpha \cdot \alpha} - \color{blue}{\frac{2 + \beta}{\frac{{\alpha}^{2}}{2 + 2 \cdot \beta}}}\right) - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

      6. unpow2 86.6%

         \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\left(2 \cdot \frac{\beta}{\alpha \cdot \alpha} - \frac{2 + \beta}{\frac{\color{blue}{\alpha \cdot \alpha}}{2 + 2 \cdot \beta}}\right) - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

   10. Simplified 86.6%

       \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\color{blue}{\left(2 \cdot \frac{\beta}{\alpha \cdot \alpha} - \frac{2 + \beta}{\frac{\alpha \cdot \alpha}{2 + 2 \cdot \beta}}\right)} - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

   if -0.99980000000000002 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 78.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 73.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 73.0%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-def 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 92.2%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 92.2%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 99.9%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/r/ 99.9%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta + \alpha}}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   5. Simplified 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\beta + \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(\left(2 \cdot \frac{\beta}{\alpha \cdot \alpha} - \frac{\beta + 2}{\frac{\alpha \cdot \alpha}{2 + \beta \cdot 2}}\right) + \left(\frac{2 + 2 \cdot i}{\alpha} + \frac{\beta + 2 \cdot i}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]

Alternative 2: 97.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := 2 + t_1\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{t_2} \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(\left(2 \cdot \frac{\beta}{\alpha \cdot \alpha} - \frac{\beta + 2}{\frac{\alpha \cdot \alpha}{2 + \beta \cdot 2}}\right) + \left(\frac{2 + 2 \cdot i}{\alpha} + \frac{t_0}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{t_0}}{t_2}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (+ 2.0 t_1)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) t_2) -0.9998)
     (/
      (+
       (/ beta alpha)
       (+
        (-
         (* 2.0 (/ beta (* alpha alpha)))
         (/ (+ beta 2.0) (/ (* alpha alpha) (+ 2.0 (* beta 2.0)))))
        (+ (/ (+ 2.0 (* 2.0 i)) alpha) (/ t_0 alpha))))
      2.0)
     (/ (+ 1.0 (/ (* (- beta alpha) (/ beta t_0)) t_2)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.9998) {
		tmp = ((beta / alpha) + (((2.0 * (beta / (alpha * alpha))) - ((beta + 2.0) / ((alpha * alpha) / (2.0 + (beta * 2.0))))) + (((2.0 + (2.0 * i)) / alpha) + (t_0 / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / t_0)) / t_2)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = beta + (2.0d0 * i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = 2.0d0 + t_1
    if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= (-0.9998d0)) then
        tmp = ((beta / alpha) + (((2.0d0 * (beta / (alpha * alpha))) - ((beta + 2.0d0) / ((alpha * alpha) / (2.0d0 + (beta * 2.0d0))))) + (((2.0d0 + (2.0d0 * i)) / alpha) + (t_0 / alpha)))) / 2.0d0
    else
        tmp = (1.0d0 + (((beta - alpha) * (beta / t_0)) / t_2)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.9998) {
		tmp = ((beta / alpha) + (((2.0 * (beta / (alpha * alpha))) - ((beta + 2.0) / ((alpha * alpha) / (2.0 + (beta * 2.0))))) + (((2.0 + (2.0 * i)) / alpha) + (t_0 / alpha)))) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / t_0)) / t_2)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = beta + (2.0 * i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = 2.0 + t_1
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.9998:
		tmp = ((beta / alpha) + (((2.0 * (beta / (alpha * alpha))) - ((beta + 2.0) / ((alpha * alpha) / (2.0 + (beta * 2.0))))) + (((2.0 + (2.0 * i)) / alpha) + (t_0 / alpha)))) / 2.0
	else:
		tmp = (1.0 + (((beta - alpha) * (beta / t_0)) / t_2)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(2.0 + t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / t_2) <= -0.9998)
		tmp = Float64(Float64(Float64(beta / alpha) + Float64(Float64(Float64(2.0 * Float64(beta / Float64(alpha * alpha))) - Float64(Float64(beta + 2.0) / Float64(Float64(alpha * alpha) / Float64(2.0 + Float64(beta * 2.0))))) + Float64(Float64(Float64(2.0 + Float64(2.0 * i)) / alpha) + Float64(t_0 / alpha)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(beta / t_0)) / t_2)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (2.0 * i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = 2.0 + t_1;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.9998)
		tmp = ((beta / alpha) + (((2.0 * (beta / (alpha * alpha))) - ((beta + 2.0) / ((alpha * alpha) / (2.0 + (beta * 2.0))))) + (((2.0 + (2.0 * i)) / alpha) + (t_0 / alpha)))) / 2.0;
	else
		tmp = (1.0 + (((beta - alpha) * (beta / t_0)) / t_2)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], -0.9998], N[(N[(N[(beta / alpha), $MachinePrecision] + N[(N[(N[(2.0 * N[(beta / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(beta + 2.0), $MachinePrecision] / N[(N[(alpha * alpha), $MachinePrecision] / N[(2.0 + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(t$95$0 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.99980000000000002

   1. Initial program 2.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 20.5%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]

   4. Taylor expanded in beta around 0 20.5%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]

   5. Taylor expanded in alpha around -inf 75.3%

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\alpha} + \left(\frac{\left(2 + 2 \cdot i\right) \cdot \left(\beta + 2 \cdot i\right)}{{\alpha}^{2}} + \frac{\left(-1 \cdot \left(2 + 2 \cdot i\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta - \left(-1 \cdot \left(2 + 2 \cdot i\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right)\right)}{{\alpha}^{2}}\right)\right) - \left(-1 \cdot \frac{2 + 2 \cdot i}{\alpha} + -1 \cdot \frac{\beta + 2 \cdot i}{\alpha}\right)}}{2} \]
   6. Step-by-step derivation

      1. associate--l+ 75.3%

         \[\leadsto \frac{\color{blue}{\frac{\beta}{\alpha} + \left(\left(\frac{\left(2 + 2 \cdot i\right) \cdot \left(\beta + 2 \cdot i\right)}{{\alpha}^{2}} + \frac{\left(-1 \cdot \left(2 + 2 \cdot i\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta - \left(-1 \cdot \left(2 + 2 \cdot i\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right)\right)}{{\alpha}^{2}}\right) - \left(-1 \cdot \frac{2 + 2 \cdot i}{\alpha} + -1 \cdot \frac{\beta + 2 \cdot i}{\alpha}\right)\right)}}{2} \]

   7. Simplified 86.6%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\alpha} + \left(\left(\frac{2 + 2 \cdot i}{\frac{\alpha \cdot \alpha}{2 \cdot i + \beta}} + \frac{\left(-\left(2 \cdot i + \beta\right)\right) - \left(2 + 2 \cdot i\right)}{\frac{\alpha \cdot \alpha}{\beta - \left(\left(-\left(2 \cdot i + \beta\right)\right) - \left(2 + 2 \cdot i\right)\right)}}\right) - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}}{2} \]

   8. Taylor expanded in i around 0 84.2%

      \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\color{blue}{\left(-1 \cdot \frac{\left(2 + \beta\right) \cdot \left(2 + 2 \cdot \beta\right)}{{\alpha}^{2}} + 2 \cdot \frac{\beta}{{\alpha}^{2}}\right)} - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]
   9. Step-by-step derivation

      1. +-commutative 84.2%

         \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\color{blue}{\left(2 \cdot \frac{\beta}{{\alpha}^{2}} + -1 \cdot \frac{\left(2 + \beta\right) \cdot \left(2 + 2 \cdot \beta\right)}{{\alpha}^{2}}\right)} - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

      2. mul-1-neg 84.2%

         \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\left(2 \cdot \frac{\beta}{{\alpha}^{2}} + \color{blue}{\left(-\frac{\left(2 + \beta\right) \cdot \left(2 + 2 \cdot \beta\right)}{{\alpha}^{2}}\right)}\right) - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

      3. unsub-neg 84.2%

         \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\color{blue}{\left(2 \cdot \frac{\beta}{{\alpha}^{2}} - \frac{\left(2 + \beta\right) \cdot \left(2 + 2 \cdot \beta\right)}{{\alpha}^{2}}\right)} - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

      4. unpow2 84.2%

         \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\left(2 \cdot \frac{\beta}{\color{blue}{\alpha \cdot \alpha}} - \frac{\left(2 + \beta\right) \cdot \left(2 + 2 \cdot \beta\right)}{{\alpha}^{2}}\right) - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

      5. associate-/l* 86.6%

         \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\left(2 \cdot \frac{\beta}{\alpha \cdot \alpha} - \color{blue}{\frac{2 + \beta}{\frac{{\alpha}^{2}}{2 + 2 \cdot \beta}}}\right) - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

      6. unpow2 86.6%

         \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\left(2 \cdot \frac{\beta}{\alpha \cdot \alpha} - \frac{2 + \beta}{\frac{\color{blue}{\alpha \cdot \alpha}}{2 + 2 \cdot \beta}}\right) - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

   10. Simplified 86.6%

       \[\leadsto \frac{\frac{\beta}{\alpha} + \left(\color{blue}{\left(2 \cdot \frac{\beta}{\alpha \cdot \alpha} - \frac{2 + \beta}{\frac{\alpha \cdot \alpha}{2 + 2 \cdot \beta}}\right)} - \left(\frac{-\left(2 \cdot i + \beta\right)}{\alpha} - \frac{2 + 2 \cdot i}{\alpha}\right)\right)}{2} \]

   if -0.99980000000000002 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 78.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 73.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 73.0%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-def 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 92.2%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 92.2%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 99.9%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/r/ 99.9%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta + \alpha}}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   5. Simplified 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\beta + \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   6. Taylor expanded in alpha around 0 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\beta}{\beta + 2 \cdot i}} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   7. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \frac{\frac{\frac{\beta}{\color{blue}{2 \cdot i + \beta}} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   8. Simplified 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\beta}{2 \cdot i + \beta}} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9998:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(\left(2 \cdot \frac{\beta}{\alpha \cdot \alpha} - \frac{\beta + 2}{\frac{\alpha \cdot \alpha}{2 + \beta \cdot 2}}\right) + \left(\frac{2 + 2 \cdot i}{\alpha} + \frac{\beta + 2 \cdot i}{\alpha}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 3: 97.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{t_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ 2.0 t_0)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.9998)
     (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)
     (/ (+ 1.0 (/ (* (- beta alpha) (/ beta (+ beta (* 2.0 i)))) t_1)) 2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.9998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / (beta + (2.0 * i)))) / t_1)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = 2.0d0 + t_0
    if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= (-0.9998d0)) then
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    else
        tmp = (1.0d0 + (((beta - alpha) * (beta / (beta + (2.0d0 * i)))) / t_1)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.9998) {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * (beta / (beta + (2.0 * i)))) / t_1)) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = 2.0 + t_0
	tmp = 0
	if ((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.9998:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	else:
		tmp = (1.0 + (((beta - alpha) * (beta / (beta + (2.0 * i)))) / t_1)) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(2.0 + t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.9998)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(beta / Float64(beta + Float64(2.0 * i)))) / t_1)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = 2.0 + t_0;
	tmp = 0.0;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.9998)
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	else
		tmp = (1.0 + (((beta - alpha) * (beta / (beta + (2.0 * i)))) / t_1)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.9998], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.99980000000000002

   1. Initial program 2.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 20.5%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]

   4. Taylor expanded in beta around 0 20.5%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]

   5. Taylor expanded in alpha around inf 86.2%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]

   if -0.99980000000000002 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 78.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 73.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 73.0%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-def 92.2%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 92.2%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 92.2%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 99.9%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/r/ 99.9%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 99.9%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta + \alpha}}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   5. Simplified 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\beta + \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   6. Taylor expanded in alpha around 0 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\beta}{\beta + 2 \cdot i}} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   7. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \frac{\frac{\frac{\beta}{\color{blue}{2 \cdot i + \beta}} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   8. Simplified 99.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\beta}{2 \cdot i + \beta}} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9998:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \frac{\beta}{\beta + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array} \]

Alternative 4: 88.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 5.2 \cdot 10^{+184}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 5.2e+184)
   (/ (+ 1.0 (/ (- beta alpha) (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.2e+184) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 5.2d+184) then
        tmp = (1.0d0 + ((beta - alpha) / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 5.2e+184) {
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 5.2e+184:
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 5.2e+184)
		tmp = Float64(Float64(1.0 + Float64(Float64(beta - alpha) / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 5.2e+184)
		tmp = (1.0 + ((beta - alpha) / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 5.2e+184], N[(N[(1.0 + N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 5.19999999999999986e184

   1. Initial program 73.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   2. Taylor expanded in i around 0 92.1%

      \[\leadsto \frac{\frac{\color{blue}{\beta - \alpha}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   if 5.19999999999999986e184 < alpha

   1. Initial program 1.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 24.8%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]

   4. Taylor expanded in beta around 0 24.8%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]

   5. Taylor expanded in alpha around inf 82.6%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 5.2 \cdot 10^{+184}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5: 87.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3.5 \cdot 10^{+185}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.5e+185)
   (/ (+ 1.0 (/ beta (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))) 2.0)
   (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.5e+185) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 3.5d+185) then
        tmp = (1.0d0 + (beta / (2.0d0 + ((alpha + beta) + (2.0d0 * i))))) / 2.0d0
    else
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.5e+185) {
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3.5e+185:
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0
	else:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3.5e+185)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(2.0 + Float64(Float64(alpha + beta) + Float64(2.0 * i))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3.5e+185)
		tmp = (1.0 + (beta / (2.0 + ((alpha + beta) + (2.0 * i))))) / 2.0;
	else
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3.5e+185], N[(N[(1.0 + N[(beta / N[(2.0 + N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.50000000000000023e185

   1. Initial program 73.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 67.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 67.0%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 83.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 83.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 83.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 83.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-def 83.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 83.8%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 83.8%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 92.8%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/r/ 92.8%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 92.8%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta + \alpha}}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   5. Simplified 92.8%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\beta + \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   6. Taylor expanded in beta around inf 91.1%

      \[\leadsto \frac{\frac{\color{blue}{\beta}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   if 3.50000000000000023e185 < alpha

   1. Initial program 1.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 24.8%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]

   4. Taylor expanded in beta around 0 24.8%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]

   5. Taylor expanded in alpha around inf 82.6%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.5 \cdot 10^{+185}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6: 81.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 1.15 \cdot 10^{+185}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 1.15e+185)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (+ (* beta 2.0) (* i 4.0))) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.15e+185) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 1.15d+185) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + ((beta * 2.0d0) + (i * 4.0d0))) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 1.15e+185) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 1.15e+185:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 1.15e+185)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(beta * 2.0) + Float64(i * 4.0))) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 1.15e+185)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + ((beta * 2.0) + (i * 4.0))) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 1.15e+185], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(N[(beta * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.1500000000000001e185

   1. Initial program 73.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 67.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 67.0%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 83.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 83.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 83.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 83.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-def 83.8%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 83.8%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 83.8%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 92.8%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/r/ 92.8%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 92.8%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta + \alpha}}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   5. Simplified 92.8%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\beta + \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   6. Taylor expanded in alpha around 0 72.7%

      \[\leadsto \frac{\color{blue}{\frac{{\beta}^{2}}{\left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)}} + 1}{2} \]
   7. Step-by-step derivation

      1. unpow2 72.7%

         \[\leadsto \frac{\frac{\color{blue}{\beta \cdot \beta}}{\left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)} + 1}{2} \]

      2. *-commutative 72.7%

         \[\leadsto \frac{\frac{\beta \cdot \beta}{\color{blue}{\left(\beta + 2 \cdot i\right) \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right)}} + 1}{2} \]

      3. +-commutative 72.7%

         \[\leadsto \frac{\frac{\beta \cdot \beta}{\color{blue}{\left(2 \cdot i + \beta\right)} \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right)} + 1}{2} \]

      4. +-commutative 72.7%

         \[\leadsto \frac{\frac{\beta \cdot \beta}{\left(2 \cdot i + \beta\right) \cdot \left(2 + \color{blue}{\left(2 \cdot i + \beta\right)}\right)} + 1}{2} \]

   8. Simplified 72.7%

      \[\leadsto \frac{\color{blue}{\frac{\beta \cdot \beta}{\left(2 \cdot i + \beta\right) \cdot \left(2 + \left(2 \cdot i + \beta\right)\right)}} + 1}{2} \]

   9. Taylor expanded in i around 0 84.4%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]
   10. Step-by-step derivation

       1. +-commutative 84.4%

          \[\leadsto \frac{\frac{\beta}{\color{blue}{\beta + 2}} + 1}{2} \]

   11. Simplified 84.4%

       \[\leadsto \frac{\color{blue}{\frac{\beta}{\beta + 2}} + 1}{2} \]

   if 1.1500000000000001e185 < alpha

   1. Initial program 1.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 24.8%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]

   4. Taylor expanded in beta around 0 24.8%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]

   5. Taylor expanded in alpha around inf 82.6%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.15 \cdot 10^{+185}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta \cdot 2 + i \cdot 4\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7: 75.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 3 \cdot 10^{+200}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3e+200)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (* i 4.0) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3e+200) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((i * 4.0) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 3d+200) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((i * 4.0d0) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3e+200) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((i * 4.0) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 3e+200:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((i * 4.0) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 3e+200)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(i * 4.0) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 3e+200)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((i * 4.0) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 3e+200], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(i * 4.0), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.99999999999999991e200

   1. Initial program 72.3%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 66.4%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 66.4%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 83.1%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 83.1%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 83.1%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 83.1%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-def 83.1%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 83.1%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 83.1%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 92.4%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/r/ 92.4%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 92.4%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta + \alpha}}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   5. Simplified 92.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\beta + \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   6. Taylor expanded in alpha around 0 72.5%

      \[\leadsto \frac{\color{blue}{\frac{{\beta}^{2}}{\left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)}} + 1}{2} \]
   7. Step-by-step derivation

      1. unpow2 72.5%

         \[\leadsto \frac{\frac{\color{blue}{\beta \cdot \beta}}{\left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)} + 1}{2} \]

      2. *-commutative 72.5%

         \[\leadsto \frac{\frac{\beta \cdot \beta}{\color{blue}{\left(\beta + 2 \cdot i\right) \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right)}} + 1}{2} \]

      3. +-commutative 72.5%

         \[\leadsto \frac{\frac{\beta \cdot \beta}{\color{blue}{\left(2 \cdot i + \beta\right)} \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right)} + 1}{2} \]

      4. +-commutative 72.5%

         \[\leadsto \frac{\frac{\beta \cdot \beta}{\left(2 \cdot i + \beta\right) \cdot \left(2 + \color{blue}{\left(2 \cdot i + \beta\right)}\right)} + 1}{2} \]

   8. Simplified 72.5%

      \[\leadsto \frac{\color{blue}{\frac{\beta \cdot \beta}{\left(2 \cdot i + \beta\right) \cdot \left(2 + \left(2 \cdot i + \beta\right)\right)}} + 1}{2} \]

   9. Taylor expanded in i around 0 84.1%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]
   10. Step-by-step derivation

       1. +-commutative 84.1%

          \[\leadsto \frac{\frac{\beta}{\color{blue}{\beta + 2}} + 1}{2} \]

   11. Simplified 84.1%

       \[\leadsto \frac{\color{blue}{\frac{\beta}{\beta + 2}} + 1}{2} \]

   if 2.99999999999999991e200 < alpha

   1. Initial program 1.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 23.1%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]

   4. Taylor expanded in beta around 0 23.1%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]

   5. Taylor expanded in alpha around inf 84.4%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]

   6. Taylor expanded in i around inf 44.8%

      \[\leadsto \frac{\frac{\color{blue}{4 \cdot i}}{\alpha}}{2} \]
   7. Step-by-step derivation

      1. *-commutative 44.8%

         \[\leadsto \frac{\frac{\color{blue}{i \cdot 4}}{\alpha}}{2} \]

   8. Simplified 44.8%

      \[\leadsto \frac{\frac{\color{blue}{i \cdot 4}}{\alpha}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3 \cdot 10^{+200}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 8: 78.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\alpha \leq 2.2 \cdot 10^{+200}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 2.2e+200)
   (/ (+ 1.0 (/ beta (+ beta 2.0))) 2.0)
   (/ (/ (+ 2.0 (* i 4.0)) alpha) 2.0)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.2e+200) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (alpha <= 2.2d+200) then
        tmp = (1.0d0 + (beta / (beta + 2.0d0))) / 2.0d0
    else
        tmp = ((2.0d0 + (i * 4.0d0)) / alpha) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 2.2e+200) {
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	} else {
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if alpha <= 2.2e+200:
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0
	else:
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (alpha <= 2.2e+200)
		tmp = Float64(Float64(1.0 + Float64(beta / Float64(beta + 2.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(2.0 + Float64(i * 4.0)) / alpha) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (alpha <= 2.2e+200)
		tmp = (1.0 + (beta / (beta + 2.0))) / 2.0;
	else
		tmp = ((2.0 + (i * 4.0)) / alpha) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[alpha, 2.2e+200], N[(N[(1.0 + N[(beta / N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(2.0 + N[(i * 4.0), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.2e200

   1. Initial program 72.3%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 66.4%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-udef 66.4%

         \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/l* 83.1%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 83.1%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      5. associate-+r+ 83.1%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      6. +-commutative 83.1%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      7. fma-def 83.1%

         \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}{\beta - \alpha}}\right)} - 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   3. Applied egg-rr 83.1%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   4. Step-by-step derivation

      1. expm1-def 83.1%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      2. expm1-log1p 92.4%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\beta + \mathsf{fma}\left(2, i, \alpha\right)}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      3. associate-/r/ 92.4%

         \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

      4. +-commutative 92.4%

         \[\leadsto \frac{\frac{\frac{\color{blue}{\beta + \alpha}}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   5. Simplified 92.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\beta + \alpha}{\beta + \mathsf{fma}\left(2, i, \alpha\right)} \cdot \left(\beta - \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]

   6. Taylor expanded in alpha around 0 72.5%

      \[\leadsto \frac{\color{blue}{\frac{{\beta}^{2}}{\left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)}} + 1}{2} \]
   7. Step-by-step derivation

      1. unpow2 72.5%

         \[\leadsto \frac{\frac{\color{blue}{\beta \cdot \beta}}{\left(2 + \left(\beta + 2 \cdot i\right)\right) \cdot \left(\beta + 2 \cdot i\right)} + 1}{2} \]

      2. *-commutative 72.5%

         \[\leadsto \frac{\frac{\beta \cdot \beta}{\color{blue}{\left(\beta + 2 \cdot i\right) \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right)}} + 1}{2} \]

      3. +-commutative 72.5%

         \[\leadsto \frac{\frac{\beta \cdot \beta}{\color{blue}{\left(2 \cdot i + \beta\right)} \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right)} + 1}{2} \]

      4. +-commutative 72.5%

         \[\leadsto \frac{\frac{\beta \cdot \beta}{\left(2 \cdot i + \beta\right) \cdot \left(2 + \color{blue}{\left(2 \cdot i + \beta\right)}\right)} + 1}{2} \]

   8. Simplified 72.5%

      \[\leadsto \frac{\color{blue}{\frac{\beta \cdot \beta}{\left(2 \cdot i + \beta\right) \cdot \left(2 + \left(2 \cdot i + \beta\right)\right)}} + 1}{2} \]

   9. Taylor expanded in i around 0 84.1%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \beta}} + 1}{2} \]
   10. Step-by-step derivation

       1. +-commutative 84.1%

          \[\leadsto \frac{\frac{\beta}{\color{blue}{\beta + 2}} + 1}{2} \]

   11. Simplified 84.1%

       \[\leadsto \frac{\color{blue}{\frac{\beta}{\beta + 2}} + 1}{2} \]

   if 2.2e200 < alpha

   1. Initial program 1.2%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 23.1%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]

   4. Taylor expanded in beta around 0 23.1%

      \[\leadsto \frac{\color{blue}{\frac{\alpha}{2 + \left(\alpha + 2 \cdot i\right)}} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2} \]

   5. Taylor expanded in alpha around inf 84.4%

      \[\leadsto \frac{\color{blue}{\frac{2 + \left(2 \cdot \beta + 4 \cdot i\right)}{\alpha}}}{2} \]

   6. Taylor expanded in beta around 0 69.9%

      \[\leadsto \frac{\color{blue}{\frac{2 + 4 \cdot i}{\alpha}}}{2} \]
   7. Step-by-step derivation

      1. *-commutative 69.9%

         \[\leadsto \frac{\frac{2 + \color{blue}{i \cdot 4}}{\alpha}}{2} \]

   8. Simplified 69.9%

      \[\leadsto \frac{\color{blue}{\frac{2 + i \cdot 4}{\alpha}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.2 \cdot 10^{+200}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \end{array} \]

Alternative 9: 72.6% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+65}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (if (<= beta 3.2e+65) 0.5 1.0))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.2e+65) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.2d+65) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.2e+65) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.2e+65:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.2e+65)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.2e+65)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 3.2e+65], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 3.20000000000000007e65

   1. Initial program 76.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]

   4. Taylor expanded in i around inf 75.4%

      \[\leadsto \frac{\color{blue}{1}}{2} \]

   if 3.20000000000000007e65 < beta

   1. Initial program 26.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. Step-by-step derivation

   3. Simplified 92.8%

      \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]

   4. Taylor expanded in beta around inf 81.1%

      \[\leadsto \frac{\color{blue}{2}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+65}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 61.6% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.5)

C

double code(double alpha, double beta, double i) {
	return 0.5;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.5d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.5;
}

Python

def code(alpha, beta, i):
	return 0.5

Julia

function code(alpha, beta, i)
	return 0.5
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.5;
end

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 63.2%

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
2. Step-by-step derivation

3. Simplified 83.5%

   \[\leadsto \color{blue}{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + \mathsf{fma}\left(2, i, 2\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} + 1}{2}} \]

4. Taylor expanded in i around inf 62.8%

   \[\leadsto \frac{\color{blue}{1}}{2} \]

5. Final simplification 62.8%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023278 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))